67 research outputs found
On diamond-free subposets of the Boolean lattice
The Boolean lattice of dimension two, also known as the diamond, consists of
four distinct elements with the following property: . A
diamond-free family in the -dimensional Boolean lattice is a subposet such
that no four elements form a diamond. Note that elements and may or may
not be related.
There is a diamond-free family in the -dimensional Boolean lattice of size
. In this paper, we prove that any
diamond-free family in the -dimensional Boolean lattice has size at most
. Furthermore, we show that the
so-called Lubell function of a diamond-free family in the -dimensional
Boolean lattice is at most , which is asymptotically best possible.Comment: 23 pages, 10 figures Accepted to Journal of Combinatorial Theory,
Series
On diamond-free subposets of the Boolean lattice
The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: A ⊂ B, C ⊂ D. A diamondfree family in the n-dimensional Boolean lattice is a subposet such that no four elements form a diamond. Note that elements B and C may or may not be related. There is a diamond-free family in the n-dimensional Boolean lattice of size (2 − o(1)) n n/2 . In this paper, we prove that any diamond-free family in the ndimensional Boolean lattice has size at most (2.25 + o(1)) n n/2 . Furthermore, we show that the so-called Lubell function of a diamond-free family in the ndimensional Boolean lattice which contains the empty set is at most 2.25 + o(1), which is asympotically best possible
Sperner type theorems with excluded subposets
Let F be a family of subsets of an n-element set. Sperner's theorem says that if there is no inclusion among the members of F then the largest family under this condition is the one containing all ⌊ frac(n, 2) ⌋-element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of F is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let P be a poset. The maximum size of a family F which does not contain P as a (not-necessarily induced) subposet is denoted by La (n, P). The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11-13, 2009], but it was somewhat updated in December 2010. © 2011 Elsevier B.V. All rights reserved
Diamond-free Families
Given a finite poset P, we consider the largest size La(n,P) of a family of
subsets of that contains no subposet P. This problem has
been studied intensively in recent years, and it is conjectured that exists for general posets P,
and, moreover, it is an integer. For let \D_k denote the -diamond
poset . We study the average number of times a random
full chain meets a -free family, called the Lubell function, and use it for
P=\D_k to determine \pi(\D_k) for infinitely many values . A stubborn
open problem is to show that \pi(\D_2)=2; here we make progress by proving
\pi(\D_2)\le 2 3/11 (if it exists).Comment: 16 page
Poset-free Families and Lubell-boundedness
Given a finite poset , we consider the largest size \lanp of a family
\F of subsets of that contains no subposet . This
continues the study of the asymptotic growth of \lanp; it has been
conjectured that for all , \pi(P):= \lim_{n\rightarrow\infty} \lanp/\nchn
exists and equals a certain integer, . While this is known to be true for
paths, and several more general families of posets, for the simple diamond
poset \D_2, the existence of frustratingly remains open. Here we
develop theory to show that exists and equals the conjectured value
for many new posets . We introduce a hierarchy of properties for
posets, each of which implies , and some implying more precise
information about \lanp. The properties relate to the Lubell function of a
family \F of subsets, which is the average number of times a random full
chain meets \F. We present an array of examples and constructions that
possess the properties
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